Algebra & Number Theory

The nef cone volume of generalized Del Pezzo surfaces

Ulrich Derenthal, Michael Joyce, and Zachariah Teitler

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We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo surface of the same degree divided by the order of the Weyl group of a simply-laced root system associated to the configuration of (2)-curves on Y. When Y is defined over an arbitrary perfect field, a similar result holds, except that the associated root system is no longer necessarily simply-laced.

Article information

Algebra Number Theory, Volume 2, Number 2 (2008), 157-182.

Received: 27 July 2007
Revised: 19 October 2007
Accepted: 11 December 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J26: Rational and ruled surfaces
Secondary: 14C20: Divisors, linear systems, invertible sheaves 14G05: Rational points

Del Pezzo surface Manin's conjecture nef cone root system


Derenthal, Ulrich; Joyce, Michael; Teitler, Zachariah. The nef cone volume of generalized Del Pezzo surfaces. Algebra Number Theory 2 (2008), no. 2, 157--182. doi:10.2140/ant.2008.2.157.

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